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Terrorism and Political Violence

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The Diffusion and Permeability of PoliticalViolence in North and West Africa

David B. Skillicorn, Olivier Walther, Christian Leuprecht & Quan Zheng

To cite this article: David B. Skillicorn, Olivier Walther, Christian Leuprecht & Quan Zheng (2019):The Diffusion and Permeability of Political Violence in North and West Africa, Terrorism andPolitical Violence, DOI: 10.1080/09546553.2019.1598388

To link to this article: https://doi.org/10.1080/09546553.2019.1598388

Published online: 20 May 2019.

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The Diffusion and Permeability of Political Violence in Northand West AfricaDavid B. Skillicorn a, Olivier Walther b, Christian Leuprecht c and Quan Zhenga

aSchool of Computing, Queen‘s University, Kingston, Ontario, Canada; bCenter for African Studies, Universityof Florida, Gainesville, Florida, USA; cPolitical Science and Economics, Royal Military College of Canada,Kingston, Ontario, Canada

ABSTRACTThis article explores the spatial and temporal diffusion of politicalviolence in North and West Africa by endeavoring to represent agroup leader's mental landscape as he contemplates strategic target-ing. We assume that this representation is a combination of thephysical and social geography of the target environment, and themental and physical cost of following a seemingly random pattern ofattacks. Focusing on the distance and time between attacks andtaking into consideration the transaction costs that state boundariesimpose, we wish to understand what constrains a group leader toattack at a location other than the one that would yield the greatestovert payoff. We leverage functional data from the Armed ConflictLocation and Event Data project (ACLED) dataset that catalogs violentextremist incidents in North and West Africa since 1997 to generatea network whose nodes are administrative regions. These nodes areconnected by edges of qualitatively different types: undirected edgesrepresenting geographic distance, undirected edges incorporatingthe costs of crossing borders, and directed edges representing con-secutive attacks by the same group. We analyze the resulting net-work using spectral embedding techniques that are able to accountfully for the different types of edges. The result is a representation ofNorth and West Africa that depicts its empirical permeability toviolence. A better understanding of how location, time, and borderscondition attacks enables planning, prepositioning, and response.

KEYWORDSPolitical violence; terrorism;borders; spectralembedding; North and WestAfrica

Introduction

The study of how crime and political violence diffuse across time and space has greatlybenefited from the increasing availability of geo-referenced data and the use of spatialstatistical analysis.1 In urban policing, for example, the design and use of hot-spot analysisbased on historical data makes it possible to anticipate when and where various kinds ofcrime are most likely to occur, and to pre-position policing assets accordingly.2 In thislimited sense, predictive modeling of crimes has been remarkably effective. The urbanenvironment lends itself to this kind of analysis: criminals are creatures of habit, they tendto travel limited distances, and some areas are naturally more target-rich than others.

There are some obvious difficulties in adapting this approach to attacks by armed non-state actors (ANSAs) in North and West Africa. As in urban settings, some natural targets

CONTACT Christian Leuprecht christian.leuprecht@rmc.ca Political Science and Economics, Royal Military Collegeof Canada, P.O. Box 17,000, Station Forces, Kingston, Ontario K7K 7B4, Canada

TERRORISM AND POLITICAL VIOLENCEhttps://doi.org/10.1080/09546553.2019.1598388

© 2019 Taylor & Francis

attract repeated attacks, for example, foreign workers in West African capitals or govern-ment forces stationed on military bases. Most victims of recent conflicts in the region are,however, civilians, killed in a rather unpredictable manner by armed groups whose mainobjective is ethnic or tribal homogeneity.3

In such wars, a multiplicity of state and non-state actors build a complex ecosystem ofaffiliated and opposing groups that also constrain when and where an attack bya particular group might occur.4 Attacks also reflect competition between traffickers andviolent extremist groups struggling to control trans-Saharan criminal networks, who oftenclash far from inhabited areas.5 Furthermore, many violent groups in the region do notlimit their attacks to a particular “turf” as urban gangs might; instead, they move relativelyfreely across the region, including across state boundaries.

Insurgent groups have limited resources and compensate by striking at locations thatmaximize impact, even abstractly via publicity, while minimizing cost. They avoid head-on confrontation, blurring the line between zones of war and zones of peace. Naval battlesare a more apt analogy. Notwithstanding strategic constraints (such as the need toblockade enemy fleets at Trafalgar and Jutland), the precise locations at which navalbattles occur are not contingent on the terrain in the way in which many engagementson land are.

This article explores the spatial and temporal diffusion of political violence in North andWest Africa. It models the strategic landscape in a group commander’s mind taking intoaccount that, far from being clinically abnormal, most violent extremists pursue collectivegoals rather than personal fantasies.6 In that sense, most violent extremists can be seen asrational actors that make choices based on costs and benefits, although their goals andactions are clearly not normal in a moral sense. The location of an attack requires a complexcalculus that combines properties of the comparative appeal of targets, the physical geo-graphy of the terrain between the current location and potential targets, the obstacles andimpediments to movement between the current location and targets, including borders thatmust be crossed, the difficulty of operating close to targets, and the need to maintain anelement of surprise. We wish to understand what motivates or constrains a group leader toattack at a location other than the one that would yield the greatest overt payoff.

The article proceeds as follows. The next section outlines existing literature on thegeographic features that are most likely to influence how attacks are conducted acrossspace and time: the distance between places, and the impediment of state boundaries.Section 3 presents the statistical properties of the attacks in this region, and how theydiffer from conventional distributions of conflict. Section 4 describes our spectral embed-ding methodology, with several extensions that allow qualitatively different similarityrelationships to be represented consistently. Section 5 shows the results of applying thismethodology to the ACLED dataset, and describes the inferences that can be drawn fromthem. The last section discusses the main implications of our work before concluding.

Networks, space, and borders

Space is now widely recognized as a fundamental dimension for ANSAs that oftenconduct operations from a territorial base, leverage geographic havens, compete withsovereign states, and fight for control over aspirational homelands.7 As a result, anincreasing number of scholars are working to integrate social network analysis and spatial

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analytical techniques.8 As Carley argues: “If we look only at the social network then thefocus of attention is on hierarchies, communication, and other social relations. Theaddition of events and locations facilitates the course of action analysis and enableslinkage to various strategic planning tools.”9

Recent conceptual and technical developments related to the spatiality of socialnetworks have primarily been applied to case studies located in the U.S., MiddleEast, Afghanistan and Pakistan, and Southeast Asia.10 The availability of large databaseshas also stimulated new approaches that use events, targets, and locations to forecast,predict or explain terrorism at the global level.11 Desmarais and Cranmer, for example,use Exponential Random Graph Models (ERGMs) to estimate the probability of newterrorist ties from one state to another between 1968 and 2002.12 Moon and Carleydevelop a multiagent model that simulates how terrorists interact with each other andwhere they relocate.13 The model shows that social networks and spatial patterns areclosely related: while terrorist networks become more geographically dispersed overtime, critical actors tend to occupy the same location. More recently, Campedelli et al.have built a model to predict future terrorist attacks based on previous targets for theperiod 1997–2016.14

By contrast, North and West Africa have received little attention from network science,despite the fact that the continent faces some of the deadliest terrorist groups in the world,including Boko Haram, Al Qaeda, Al Shabaab, and the Islamic State. Most of the studiesconducted so far have apprehended the spatiality of social networks based on actors whoselocation or territory was well known, whether in conflict studies or internationalrelations.15

This article seeks to address territories where fixed targets are the exception, focusingon the effect of two fundamental geographic features: distance and borders. Distanceconstrains locations of attacks in two ways. First, when attacks involve the same people orresources, these must be transported from one location to another, which takes time andcosts money. Second, a distant location imposes transaction costs: unfamiliarity with thephysical and social terrain, different languages, and so on.

Borders are one important aspect of the effect of distance.16 As Engel and Rogers andBorraz et al. showed, borders introduce price distortions that are equivalent to adding anextra distance between locations.17 Borders also limit social exchanges—even when peopleuse social media—and are a major impediment to labor market integration, despite formalagreements that promote the mobility of labor.18 In addition to hindering the mobility ofgoods and people, borders also have strong effects on political violence, which oftenoccurs at the subnational level.19

Even when borders offer jurisdictional protection and opportunities to create safehavens, they nonetheless distort distance and affect the mobility of armed groups.20 Theeffect of borders on the decision by a group to carry out attacks in more than one countrycan, therefore, be modeled as obstacles to be surmounted. The practical cause of theobstacle might be the overhead of the crossing, either overtly or covertly; differences inculture on the other side; or the increased risk associated with operating away from “hometurf,” where, for example, it may be less obvious who can or should be bribed. (Otherkinds of border-like effects exist, based on differences in ethnicity and language, and thesecould be integrated into our approach, but the data is harder to obtain at scale).

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This article proposes a novel approach to modeling the impact of borders: as their effectincreases, the perceived distance structure becomes less and less planar, and so a simplemap representation becomes less and less accurate – borders behave like mountain rangeson the surface of the earth, making some movements harder and funneling others in non-obvious directions. The modification we suggest is to represent the locations where attackshave taken place as a network where the nodes are attacked locations and the edgesrepresent perceived distances (increasingly modified) between them. This network will notbe planar, because perceived distances will differ from physical distances, but our model-ing technique allows them to be mapped back into two dimensions for visualization.

The uneven geography of attacks in North and West Africa

As we would expect, statistics show that the distribution of the 29,272 attacks by 921groups at 1831 locations from 1997 to 2015 is not random. The location where the mostattacks took place is near Benghazi in Libya, where 1230 attacks are recorded. However,the mean number of attacks per location is sixteen, and the mean number of attacks forthe least-attacked 1600 locations is only 4.7; so, the distribution is highly skewed.Histograms of the attack frequencies are shown in Figure 1. If we consider instead howmany organizations have carried out attacks at each location, the highest score isa location near Tripoli where sixty different groups have carried out an attack.However, the mean number of groups attacking at a given location is 3.7; so, again, thedistribution is skewed. Of course, these highly skewed distributions mean that conven-tional hot-spot analysis can, and should, be carried out. However, in the North and WestAfrica setting it cannot be enough, because of the constantly shifting set of actors,allegiances, and motivations.

These observations make it clear that this setting is far removed from conventionalwarfare where the ground is taken and held; and that participants do not form large, stableblocs. Rather, interactions are fluid and consist of smaller, constantly shifting membersand alliances.21 The decision about where to carry out an attack is constrained by two setsof factors: properties of the target, and properties of the attacking group. Properties of the

Figure 1. Frequency histogram of the locations of all attacks (left) and the least-frequent 1600 attacks,showing how skewed the distribution is.

4 D. B. SKILLICORN ET AL.

target can be used to analyze risk; targets are attacked for a reason, and their appeal at anygiven moment can be assessed. The more difficult properties are those of the attackinggroups, whose internal processes may be opaque, and who are fundamentally motivated todo the unexpected. However, such groups cannot attack at will – they are constrained byresources, broadly interpreted. We focus on the constraints imposed by distance (whichmatter in a non-urban environment) and the way borders distort distance.

Figure 2 shows the distribution of all attacks by latitude and longitude from 1997 to2015. The main clusters of violence, in decreasing order of fatalities reported in theACLED data, are located in Nigeria, Northern Algeria, Northern Libya, the Chad-Sudanborder, and along the Gulf of Guinea. Nigeria is especially affected, with 50,144 fatalities,most of them resulting in either from ethnic violence, fights to control oil production inthe Niger Delta, or from attacks by Boko Haram. In West Africa, the border between Chadand Sudan remains a focus of conflict due to persistent fighting between the Sudanesegovernment and rebels in Darfur. The portion of the Gulf of Guinea that extends fromAbidjan to Banjul has suffered from a succession of civil wars in Ivory Coast, Liberia,Sierra Leone, and Guinea-Bissau.

In North Africa, Algeria has also been markedly affected by violence, principally due toactivity by three organizations in conflict with the Algerian government: the ArmedIslamic Group (GIA), the Salafist Group for Preaching and Combat (GSPC), and AlQaeda in the Islamic Maghreb (AQIM). Violent Islamist groups were involved in 93percent of the 12,050 fatalities in Algeria. With 12,610 fatalities reported, Libya is the thirdepicenter of violence, principally because of the overall political instability after the ousterof Colonel Gaddafi in 2011 and the subsequent civil war. In comparison, the Sahel and

Figure 2. The positions of all attacks by latitude and longitude. The Mediterranean coast can be seen atthe top of the figure, the Atlantic coast on the left and the Gulf of Guinea on the lower side.

TERRORISM AND POLITICAL VIOLENCE 5

Sahara regions are less immediately affected by violence, with the notable exception ofnorthern Mali where secessionist rebels and Islamist groups have opposed the governmentsince 2012. More than 1,200 of the 2,761 victims of violent events reported in Mali from1997 to 2015 died in an event involving one or several Islamist groups, including AQIM,Ansar Dine, the Movement for Oneness and Jihad in West Africa (MUJAO) and AlMourabitoun. In Mauritania, where violent Islamist groups have also been active, thenumber of victims resulting from clashes with such groups is much lower, with eighty sixfatalities, while in Niger, the number of victims (997) has increased rapidly due to BokoHaram.

Often organizations from one country carried out attacks in another. For example, theIslamist group AQIM, historically based in Algeria, has conducted numerous attacks inneighboring countries. Boko Haram is also responsible for attacking civilians and securityforces in Niger, Chad, and Cameroon. The transnational activity of ANSAs has promptedmany governments of the region to carry out attacks abroad. At the beginning of the2010s, for example, the Mauritanian military carried out attacks in Mali to destroy militarybases belonging to AQIM. More recently, Chad sent troops to both Nigeria and Cameroonto fight Boko Haram.

Methodology

Spectral embedding

While networks are a natural way to represent relationship data involving nodes andconnection patterns, the conventional adjacency matrix representation is difficult tounderstand. Two approaches to making network data intelligible are used: graph drawing,and graph embedding. Graph drawing, on the one hand, attempts to provide the mostunderstandable visualization of a network by placing the nodes and edges so that they donot occlude one another, while still placing nodes that are similar as close to one anotheras possible. It emphasizes clarity in the resulting picture. Graph embedding, on the otherhand, tries to represent a network as accurately as possible, that is so that the distancesbetween each pair of nodes are as representative of their similarity in the network aspossible, at the expense of producing a picture that may be hard to understand directly.Graph drawing produces a representation that is qualitatively accurate, while graphembedding produces a representation that is quantitatively accurate.

We will use spectral embedding, the most effective of the graph embedding approaches,to represent networks in a geometric form.22 This requires a mathematical technique withtwo main steps.

First, the adjacency matrix, A (of distances, say) is converted to one of a family ofLaplacian matrices. We begin by using the combinatorial Laplacian, L, given by the matrixequation: L = D – A, where D is the matrix whose ith diagonal entry is the total edgeweight of the edges connected to node i, with all of its other entries zeros. Since A issymmetric, so is L. In other words, the diagonal entries of L are the (weighted) degrees ofeach node, and the off-diagonal entries are the negatives of the corresponding edgeweights in the adjacency matrix.

6 D. B. SKILLICORN ET AL.

Second, an eigendecomposition of L is computed such that: L = Q Λ Q’, where Λ isa diagonal matrix of eigenvalues, in decreasing order, Q is the n × n eigendecompositionof L, and the superscript dash indicated matrix transposition.

If the network is connected, then the final eigenvalue is zero, and we ignore the finalrow of Q. The k rows preceding it can be interpreted as the coordinates of each node ofthe graph in a k-dimensional space. In other words, if we take the n rows of Q, and the(n-1)st and (n-2)nd rows of Q we can use these as coordinates to place points correspond-ing to the n nodes in a two-dimensional rendering of the network.

This spectral embedding comes with strong mathematical guarantees that it is the mostfaithful representation of the network structure in the chosen number of dimensions.23

From an intuitive viewpoint, the effect of spectral embedding is to begin with the cloud ofpoints representing the nodes of the graph in a space of dimension n-1, in which thesimilarities (distances) between each pair can be represented exactly. The eigenvectors arethen oriented along with orthogonal directions for which the cloud has the greatestvariation: the (n-1)st eigenvector along the direction of greatest variation, the (n-2)ndalong a direction orthogonal to this with the next greatest variation, and so on. Choosingk = 2 creates a projection of the graph into the two directions that most accuratelyrepresent the distances between every pair of nodes, given the constraints from all ofthe other nodes. Spectral graph embedding, as the name suggests, has connections to theLaplace-Beltrami operator, and the eigenvectors produced can be regarded as capturingvibrational modes of a graph. Thus, it generalizes forms of graph analysis based ondifferential and partial differential equations.

We apply this spectral approach to a network of attack locations. We want a network inwhich edge weights are large for nodes that are strongly connected and small for nodesthat are weakly connected. Distances, however, are the exact opposite – locations that arefar apart (and so weakly associated) have large values for their mutual distance. The edgeweights need to be inverted so that close locations have large weights and vice versa. Thereare several ways to do this, but we choose to subtract each distance from the long-est distance between any pair of locations in the network increased slightly by multiplyingit by a factor of 1.1.

Extending the spectral embedding to represent multiple relationships

We want to construct a network in which both distances and border crossing difficulty arerepresented consistently. The spectral approach can be extended to represent two differentsimilarity properties with a new, enhanced network. This network representation is builtas follows: each node is replicated into two versions, a red version, and a green version.The red versions of the nodes are connected using the adjacency matrix based on onesimilarity property (distances) and the green versions are connected using the adjacencymatrix based on the other similarity property (border crossing difficulty). We can think ofthese two subgraphs as forming two layers. Now we connect each of the pairs of replicatednodes by new (say, blue) edges. Thus, we have a single graph with red and green nodes,and red, green, and blue edges.

The question is now how to assign weights to the new, blue edges. The larger theseweights are, the more the two layers are forced to be “aligned.” We can imagine that, inthe larger graph, edges behave like springs that pull the nodes they connect with a force

TERRORISM AND POLITICAL VIOLENCE 7

proportional to their edge weight. In each layer, nodes are pulled together based on theircloseness (in the red layer) or cross-border accessibility (in the green layer), but they arealso pulled together by how consistent their “role” is in both layers at once. Note that theedge weights in the distance layer are much larger than those in the border layer and anadjustment must be made to compensate for this.

To see how to assign weights to the blue edges, we convert the adjacency matrices torandom walk matrices by computing the sum of edge weights in each row, and dividingthe entries of each row by its sum. The matrix entries are now values between 0 and 1,which can now be interpreted as probabilities. The name “random walk” comes fromimagining a walker who inhabits the network, constantly moving from node to node alongthe edges. At any given node, the walker chooses which node to visit next by choosingamong the outgoing edges in proportion to their probabilities. The random-walker view ofa network is quite elegant: for example, the proportion of time that a random walkerspends at each node is a measure of its importance, since important nodes tend to be wellconnected and so are easy to visit regularly.

A variation of a random walk matrix, with better properties, is a lazy random walkmatrix. Here the entries of each row are divided by twice the sum of the row, so that theentries sum to 0.5. The remaining 0.5 is placed in the diagonal position. The interpretationis now that the random walker makes decisions among the outgoing edges as before, butmay choose, with probability 0.5, to remain at the current node for the next step.

We use this as a model to motivate the new 2n × 2n random walk matrix. The entriescorresponding to each subgraph are mapped to values between 0 and 0.5, the diagonals areleft as zeros, and the remaining 0.5 probability is assigned to the blue edges between thelayers. Thus, from a random walk perspective, a lazy random walker behaves in each layeras it would before, but can move from layer to layer with probability 0.5 on each step.

The random walk matrix for the larger graph is bigger (2n × 2n) but most of the extraentries in this matrix are zeros. The top left-hand corner is the random walk matrix fromthe distances, the lower right-hand corner is the random walk matrix of border crossingpermeability, and the other two corners are diagonal matrices of the edge weights of theblue edges, and so mostly zeros. The cost of computing an eigendecomposition dependson the number of non-zero values in the matrix, so computing an eigendecomposition forthe bigger graph is not much more expensive than computing one for each of the existingnetworks separately.

Now the standard spectral embedding algorithm can be used to convert this largeradjacency matrix to a Laplacian, compute its eigendecomposition, and embed the resultinggraph in a two-dimensional space. In this random-walk embedding, each geographicallocation is represented by embedded red and green points. The distance between the twopoints corresponding to the same location, the length of the embedded blue edge betweenthem, reflects how different their roles are from the perspective of distance and theperspective of borders. Locations for which these points are far apart are of particularinterest.

Extending the spectral embedding to represent a third property

We also want to be able to model the sequential patterns of attacks. We do this byenriching our representation with a third layer, in which two nodes are connected by

8 D. B. SKILLICORN ET AL.

a directed edge when an attack at one is followed immediately by an attack at the other.The fact that the edges are directed in the new third layer introduces three complicationsto the embedding process. First, in an undirected network, the importance of a node isproportional to the total edge weight of the edges connected to it. This is no longer truefor a directed network – a node may have many heavily weighted incoming edges, but itmay not be important if these upstream nodes themselves are hard to reach. Importancederives from the entire graph, rather than being a local property. Second, a random walkercan become trapped at a single node that only has incoming edges (but this is easy todetect), or in a region that collectively has no outgoing edges (and this is expensive todetect). Third, a random walk matrix is not necessarily symmetric, which the standardembedding algorithm requires.

A solution to these problems was developed by Chung,24 but it has a number ofdrawbacks. Instead, we use a newly developed approach that models edge direction byreplicating each node into an outgoing version and an incoming version, and connectingthese in the obvious way by undirected edges. This reduces the problem of embedding anundirected graph at the expense of adding more layers. As before, new edges have to beadded between the versions of the same nodes.25 This approach has been validated againstthe known social structures of the Florentine families in the time of the Medicis,26 and hasbeen applied to understand the structure of criminal networks.27

The construction is as follows: we combine three layers, the geodesic-distance layer,the border crossing difficulty layer, and the time-sequence layer. In the previousconstruction, we converted each layer into a random walk matrix, divided the totaledge weights incident at each node in half, and allocated half to edges that remain inthe layer (proportional to their original values) and a half to the edge to the other layer.We cannot follow this strategy for the three-layer graph because the directed adjacencymatrix cannot be converted to a random walk matrix. However, we use the sameintuition: the total outgoing edge weight incident at each node should be divided inhalf, with half remaining within the layer, and half allocated to the edges to otherlayers. There are now two other layers, so the amount allocated to other layers is splitequally between them.

The edge weights in the different layers are of considerably different magnitudes. Tocompensate we must normalize each subgraph adjacency matrix to make the magnitudesof the edge weights comparable between the layers. We do this by dividing the entries foreach layer by the mean of the non-zero entries in that layer. Almost all non-zero values areclose to 1, with the exception of larger entries in the sequence layer.

A further complication arises because the sequence layer is extremely sparse; incomparison, the other two layers are fully connected. If the sequence layer was to beembedded by itself, there would be two strong clusters with a weak connection betweenthem, and many isolated nodes. The isolated nodes would be embedded at the origin, withthe two clusters in a dumbbell shape. When this layer is connected into the larger graph,the other two layers have the effect of connecting all of the nodes in the sequence layer toone another indirectly by paths of small weights. Nodes with no connections within thesequence layer are embedded close to the center, nodes with weak connections within thesequence layer are embedded further out, and nodes that are strongly connected withinthe sequence layer are embedded furthest out. This is, of course, the inverse of what wewould want—important nodes being embedded centrally. The solution is to normalize the

TERRORISM AND POLITICAL VIOLENCE 9

edge weights in the sequence layer further by making the sum of the edge weights incidentto each node constant—so that nodes without any connections in the layer are givena heavily weighted self-loop, nodes with weaker connections in the layer a less-weightedself-loop, and so on. The result is a three-layer structure, with added directed edgesbetween copies of the same node in the different layers.

We now replicate each layer into two more sublayers, one to which the outgoing edgesare attached, and one to which the incoming edges are attached. All edges are nowundirected; so, we can use the standard embedding (on a 6n × 6n matrix whose entriesare almost all zeros). The newly replicated copies must also be connected to one anotherby edges whose weights depend on the incident edge weights of the pair. This constructionis intricate but essentially straightforward.28

Applying spectral embedding to attack location data

The Armed Conflict Location and Event Data project (ACLED) dataset catalogs violentextremist incidents in North and West Africa since 1997. Rich data for each incident isavailable, including timing, groups participating as attackers and victims/targets, andlocation, both in terms of latitude and longitude, and by administrative district.29 Werestrict our attention to incidents that are clearly categorized as violent and fall into one ofthe following categories: battle with and without change of territory; riots and protests;violence against civilians; and remote violence. Attack locations for our purposes are at thegranularity of local administrative districts.

We use this data to generate a form of “social networks”. As opposed to a conventionalsocial network where humans are nodes and relationships are edges, in this article nodesare abstractions of locations, and edges represent distances as perceived by the actorsconcerned. In the resulting “social network,” then, nodes are administrative regions, anapproach similar to the one described by Batagelj et al.,30 with edges between them thatare of qualitatively different types: one set of undirected edges representing geographicdistance, another set of undirected edges representing the costs associated with having tocross borders, and a set of directed edges representing consecutive attacks by the samegroup at two locations. This network represents the landscape as perceived by groupleaders, balancing their perceptions of the next potential attack at a “remote” location(with an element of surprise) against the convenience of choosing instead a close, evena repeat, location.

We analyze the resulting network using spectral embedding techniques that combinethese different types of edges into a representation of North and West Africa that depictsits empirical permeability to attacks from the perspective of any violent group. Whendistributions are highly skewed, as they are in this setting, statistical measures such asaverages are useless for planning effective counterinsurgency deployment. This map ofpermeability reflects the impact of distance, borders, and time on violent group actions,and so provides a first step towards principled planning, prepositioning, and response.

Modeling distances between attacks

We begin our modeling with a network derived from the location data by building an n ×n adjacency matrix, A, with rows and columns corresponding to locations, and whose ijth

10 D. B. SKILLICORN ET AL.

entry is the weight of the edge connecting them. If the edges are undirected, then the ijthand jith entries are the same, and the matrix is symmetric. The matrix has 1831 nodesrepresenting the attack locations, fully connected by edges whose edge weights are thegeodesic (great circle) physical distances between them. These distances were calculatedfrom the latitudes and longitudes using the Haversine function.

The key functional property of spectral embedding is that it allows an arbitrary networkto be represented in two dimensions in such a way that the distance between any pair ofnodes accurately represents their dissimilarity. This is needed even for the simple networkthat connects locations where attacks have taken place, because these locations exist on thecurved surface of the earth (which matters at this scale) and so must already be trans-formed to produce a flat two-dimensional representation.

In the figures that follow, the nodes of the network are color-coded by the countries inwhich they are located. The top left map on Figure 3 shows an embedding of the attacklocations based purely on the geodesic distance between all pairs. The difference betweenthis and Figure 2 (based on position) is that the network of location similarity drawslocations closer together when attacks happen in closer proximity. In other words, hot-spots get hotter; or, from the perspective of group leaders, attacks in close proximity toprevious attacks are, empirically, a popular option. They are more likely to take place closetogether than logistic considerations or a desire for surprise would indicate.

Figure 3. Spectral embedding based on geodesic distance, and then with borders modelled asequivalent to distances of 50 km, 100 km, 500 km.

TERRORISM AND POLITICAL VIOLENCE 11

National borders can be seen when the locations are color-coded by country, but wouldbe less obvious if they were not. This indicates that there are few differences in thelocations of attacks along the Gulf of Guinea, or along the Mediterranean coast; butBurkina Faso and Mali show clear separations from their southern neighbors; and there isa strong separation between attacks in the Mediterranean countries and those farthersouth. Mauritania, and to some extent Algeria, bridge these two regions.

The effect of linear (additive) border costs

We now examine the effect of the presence of borders as it might enter into thecalculus of a group planning its next attack. A simple way to model the cost of crossinga border is as an increased distance between origin and destination. For example, theaddition of 100 km to account for crossing a border captures a delay of, say, two hours(assuming typical speeds of 50 km/h for travel using local pick-up trucks) caused by theoverheads of the crossing.

We compute the number of borders that must be crossed to pass between all pairs ofthe twenty-one countries we consider. This calculation was done based on the great circledistance between a median point in each country but when such a route would haverequired crossing many borders and a slightly longer route would have required crossingmany fewer, the lower number of border crossings were used. For example, a direct pathfrom Sierra Leone to Niger passes through Guinea, Ivory Coast, and Burkina Faso, buta path through Guinea and Mali crosses fewer borders without adding much actualdistance. Of course, some borders are less permeable than others due to government ormilitary policy. This can be straightforwardly modeled by altering the effective addeddistance for each border.

In a model where border crossings are modeled as artificial added distances, the effectof multiple crossings is linear, since crossing two borders is twice as expensive as crossingone. Given a distance equivalent for each border crossed, we add this distance to the edgeweight associated with each pair of nodes before inverting distances as described earlier.Additive border crossing can be modeled in a single network by this adjustment of theedge weights.

Figure 3 shows the embedding when borders are modeled as equivalent to an increaseddistance of 50 km between countries. The maximum distance between attack locations inthe dataset is almost 5000 km, so this is a small distortion, and indeed the differencesbetween Figure 3 are too small to see at this resolution. However, when this distortion isincreased to 100 km, the situation changes. On the one hand, attack locations in differentcountries now begin to separate in the visualization, indicating that they have become lesssimilar, especially along the Gulf of Guinea. On the other hand, the border betweenAlgeria and Tunisia shows little change, indicating how similar attack locations in thesecountries are. When the effect of a border is increased to be equivalent to 500 km, Figure 3shows that locations clearly separate by country. When the cost of a border crossing is asgreat as this, cross-border locations seem less similar, and locations within the samecountry, by contrast, seem more similar to one another.

From the perspective of a group leader at a particular location and considering thelocation for a next attack, these results suggest that the presence of a border has littleimpact until the potential overhead of crossing that border is at least equivalent to the

12 D. B. SKILLICORN ET AL.

costs of 100 km of intra-country travel. This has implications for the amount of efforta country should put into hardening its border to have any effect on the attack calculus ofANSAs.

Non-linear border costs

While an argument could be made for linear border costs, it seems more plausible that theperceived cost of crossing borders is non-linear. For example, suppose that the probability ofinterdiction at any given border is 20 percent. Then, the probability of interdiction whencrossing two borders is 36 percent, since there is an 80 percent chance of successfully crossingthe first border, and an 80 percent chance of success at the second border, so the probability ofcrossing both successfully in the sequence is 0.8 × 0.8 = 0.64. Thus, a group planning an attacktwo countries away should perceive it as substantially more costly than one in a neighboringcountry.

There are arguably two ways in which groups might frame non-linear border costs. On theone hand, a group with pan-national ambitions, such as AQIM, must exert its influence bycarrying out attacks in countries that are far away from its center of influence. For suchgroups, a cost of crossing borders might be appropriately framed in terms of the rate ofsuccess, and values as large as 95 percent would be necessary for them to succeed acrossmultiple borders. On the other hand, a group whose interests are primarily domestic, such asBoko Haram, might regard borders as substantial impediments to their choice of locations toattack, both because of the discomfort of operating in another country, and the reducedimpact such an attack might have on their local agenda. For such groups, a much lowersuccess rate associated with crossing borders, perhaps 50 percent, might appropriately frametheir calculus.

A new border adjacency matrix for the 1831 locations was computed by setting the ijthentry to a given border success probability (between 0 and 1) raised to the power of thenumber of borders between the country of location i and the country of location j. If theprobability of success is 0.95, then the result is 0.95°= 1 for locations in the same country,0.95 for locations in neighboring countries, 0.952 = 0.9 for locations two countries apart,0.953 = 0.86 for locations three countries apart, and so on. If the probability of success isonly slightly smaller, the effect becomes more pronounced. For a probability of success of0.9 per border, the rate of success for crossing two borders sequentially is 0.92= 0.81 andfor three, 0.93 = 0.73.

Borders as a separate layer

Non-linear costs of borders cannot be represented as an addition to the representation ofdistances, because their impact depends on the particular pairs of locations being con-sidered. Instead, we consider locations to be connected by two kinds of relationships: theobvious one based on how far apart they are, and the other by how many borders must becrossed on the path between them. Thus, there are two networks, with the same set ofnodes, but qualitatively different types of edges. We use the two-layer approach to builda single network by connecting the two networks by adding an extra edge (“blue”)between the two versions of each nodes, and adjusting the edges weights as describedabove. The resulting network can then be embedded using a spectral technique. Each

TERRORISM AND POLITICAL VIOLENCE 13

location has two positions in the resulting two-dimensional embedding, reflecting theirroles from distance and border perspectives. The distance between these two positionsshows how much impact borders have on their perceived distance from other locations.

Figure 4 shows the full embedding of the two-layer graph with border-crossing successprobability set to 95 percent. The red versions of the nodes, which capture the embeddedlocations based on geography, lie around the outside of the embedding, being “pushed”apart by the effect of borders; the green versions of the nodes, which capture theembedded locations based on border crossing difficulty, lie further towards the center,pulled inwards by a relatively smaller effect of border crossing costs; and the blue linesindicate the magnitude and direction of the difference for each location.

Figure 5 shows the embedding of the locations based on distance, which represents our“red” nodes color-coded by the countries in which they are located, as before. Comparingthis figure to Figure 3, where borders were represented as equivalent to distances of100 km, shows that the spread of locations is not dissimilar – but there is a greater spreadfrom east to west, as expected given the number of borders along the Gulf of Guinea.

Figure 6 shows the difference between locations based on distance and based on bordercrossings. The “blue” lines, also color-coded by country, make it clear that the effect ofborders is effectively to spread locations further apart from a virtual center in SouthernAlgeria, a fixed point where the distance to all other locations in North and West Africa isproportional to the number of borders that have to be crossed to reach them. This pointcorresponds to the commune of Bordj Badji Mokhtar, in Adrar Province, Algeria. The

Figure 4. Embedding of two-layer graph (red – locations based on distance, green – locations based onborder crossings, blue lines – difference between the two) with border crossing probability 0.95.Borders push locations apart by amounts that depend on the global similarities.

14 D. B. SKILLICORN ET AL.

embedding shows that Bordj Badji is the most central place to use as a base when bordercrossing costs, as well as distance, are taken into account. Indeed, Bordj Badji Mokhtarand the adjacent trading town of Al-Khalil in Mali have long been known for beinga haven for arm and drug traffickers and a central node in the transnational network thatconnects local Tuareg tribes with the Algerian military and secret police.31 Between 2008and 2014, Bordj Badji Mokhtar saw repeated clashes between Algerian governmentalforces and Islamist groups such as AQIM and MUJAO.

Figure 6 shows a similar figure but with the border-crossing success rate reduced to 80percent. With this assumption, the distortion introduced by borders is quite different.Locations in the north and center of the region show the same radial distortion, in whichlocations appear further apart than they are because of the presence of borders. But, forthe countries in the southwest and southeast, the distortion introduced by borders isoriented orthogonally to the previous distortion. For example, Sierra Leone and Liberia“push” one another apart rather than being influenced by distant Algeria and Tunisia; andNigeria and Cameroon show a similar pattern.

Figure 6 shows what happens when the probability of successfully crossing borders isreduced to 50 percent, reflecting the mindset of groups with primarily local agendas.Distortions caused by borders are almost completely local, depending primarily on a fewnear neighbors. Because of the roughly triangular shape of North and West Africa, the neteffect is that most distortions align toward the center but, since the probability of crossinga substantial number of borders drops quickly to a small value, countries are only weaklyconnected. Note that Cameroon sees the rest of the region through the lens of Nigeria,from which it is now indistinguishable. This situation reflects the increasing interdepen-dence between the two countries since Boko Haram, historically active in Nigeria, spreadto adjacent countries in 2014.

Figure 5. Spectral embedding based on non-linear border costs, with border crossing probability 0.95.

TERRORISM AND POLITICAL VIOLENCE 15

Time sequence as a separate layer

So far, we have ignored the dimension of time. Consider the sequence of attacks by eachparticular group. Unless subgroups of the group act completely independently, there aresome constraints on the sequencing of attacks, and these may offer insights into thegroup’s constraints or strategy. For example, if the same (sub)group carries out successiveattacks, perpetrators must travel from one location to another, perhaps taking or gatheringmaterial; and perhaps crossing borders as well. Even if the attacks are not carried out bythe same individuals, the time sequence reflects, at some level, strategic thinking by thegroup’s leadership.

From the ACLED dataset, we extract a third adjacency matrix connecting successiveattacks by the same group by a directed edge of weight 1. If the ijth entry is 1, then the jithentry is unlikely to be, so this matrix is typically asymmetric.

Our scope includes all the armed groups for which a clear transnational activity isdocumented in the ACLED from 1997 to 2015. This includes Boko Haram and nine otherIslamist groups affiliated with Al Qaeda that share a common historical and ideologicalbackground and form several components of a single, flexible network: Al Qaeda, Ansare

Figure 6. Difference between embedded positions based on distance and border crossings: crossingprobability 0.95, 0.80, 0.50.

16 D. B. SKILLICORN ET AL.

Dine, AQIM, GIA, Al Mourabitoune, GSL, GSPC, MUJAO, and Those Who Signed inBlood.32 Note, however, that AQIM has historically occupied two very distant regions: theKabylie region in northern Algeria, where its “national” emir Abdelmalek Droukdelsupposedly resides, and the Sahel-Sahara region where several sections (katibas) havedeveloped since the mid-2000s.33 Only the events related to the Sahelian-Saharan sectionof AQIM were considered. Mapping the chronological activity of both AQIM’s Kabylianleadership and Sahelian-Saharan sections would give the impression that much of theattacks span the Saharan desert, which is probably unrealistic. To solve this issue, weconsider separately all violent events south of In Amenas, Algeria. Because most of thesegroups have a trans-national agenda, we use a border crossing probability of 95 percent inthe analysis.

Figure 7 shows the locations’ embedded distance as in Figure 3, but overlaid by blackedges connecting sequential attacks by the same organization (that is, whenever there isa one in the newly constructed directed adjacency matrix). The map confirms the pan-regional ambition of the nine Islamist groups affiliated with Al Qaeda that conductedattacks from Mauritania to Chad, often across borders. Their patterns of attack divergegreatly from those of Boko Haram, the majority of whose violent attacks took place withinNigeria itself.34 It is also notable how large the distances between successive attacks can be.

We now want to extend the embedding to include the time-sequence structure, that isto incorporate the empirical similarity of each pair of sequential attacks. We do this byextending the layered model to three layers: one representing geodesic closeness, onerepresenting border permeability, and one representing sequence in time.

Figure 7. Spectral embedding as in Figure 3, overlaid by lines connecting sequential attacks by thesame group (for 9 groups with trans-national intentions, and Boko Haram).

TERRORISM AND POLITICAL VIOLENCE 17

The resulting embedding is shown in Figure 8: Boko Haram attacks are local toNigeria, with occasional incursions into Chad, Cameroon, and Niger. Several locationsare subject to repeated attacks. The attacks by Islamist insurgents affiliated with Al-Qaeda are concentrated in northern Algeria, but another nexus of attacks can beobserved in southern Algeria, Mali, Mauritania, and Niger. These second groups ofattacks are much more widespread goegraphically. Note that, as locations attackedsequentially appear more similar, locations where this does not happen spread furtherapart in the embedding.

Conclusion

The aim of this article is to provide a dynamic description of the structure of conflict inNorth and West Africa across time. While the causal inferences that can be drawn arenecessarily limited, our findings significantly advance the state of knowledge in networkscience and conflict studies.

First, the article expands our understanding of the way structural data can be extendedinto the analysis of conflict through the application of spectral embedding techniques tonetwork science. We have shown how newly developed extensions to spectral embeddingtechniques (with typed edges, and directed edges) that have been previously applied toconventional social networks, where humans are nodes and relationships are edges, can beextended to social networks in which the nodes are locations and edges represent distancesas perceived by the actors concerned.

Figure 8. Spectral embedding of the 3-layer network, with locations attacked sequentially appearingmore similar. Note the difference from Figure 7, where sequence plays no role in embedded position.

18 D. B. SKILLICORN ET AL.

Second, the article suggests ways to think differently about the nexus of space andconflict: commonly we think of networks as a function of place; instead, this article invertsthis convention to think of the place as a function of networks. Place is no longer simplya physical location—its positional and strategic importance define it, and these changes asthe movements that give rise to networks change. Our study of political violence in Africais an application of ideas from social networks to networks of a different kind, reflectingthe fact that place is a human construct as well as a physical one.

An interesting finding of our work is to show that some of the most violent places inthe region are far from inhabited areas, such as the extreme north of Mali or the north-eastern reaches of Niger. Instead of thinking of conflicts as a function of place per se, wecan now think of conflicts as a function of movement. Since movement, unlike place, isnot fixed, strategic consideration can now be given to ways to influence, alter, or dispersesome movements while generating and encouraging others.35 Conflict has long beenknown to be dynamic; this article posits a method to model that dynamism, one thatmakes it possible to respond to conflict and violence in terms of strategic consideration ofmovement rather than simple spatial coordinates.

Finally, the spectral embedding of these networks of places provides an insight into thepotential mindset of group commanders as they consider their next actions. Geodesicdistance obviously plays a role, but other constraints are also significant. Borders are onesuch constraint. Future work might explore the effect of more complex factors as politicalconditions in an adjoining country, the type of regime, and state-sponsorship of armedgroups on the model and results. The aim of this article, however, is limited to two effects:geodesic distance and borders. Borders are modeled from two perspectives: that of a groupwith pan-national aspirations, and that of a group with more local aspirations, and shownthat the mental landscapes produced differ. In other words, we show that intent, cap-ability, and opportunity affects the framing of the transaction costs imposed by crossingborders and resource constraints of operating across vaster distances, and so affect thecalculus of a group’s leadership. Constraints as simple as habits also play a role: at somepairs of locations attacks occurred at the same pair in sequence sixteen times.

Our findings have implications for conflict prevention and early intervention in theregion. For example, consider the practical problem of anticipating, after an attack bya particular group, the probable timing, and location of the next attack. Without extrinsicinformation, the conventional approach is simply to draw concentric circles around thelocation of the current attack, and assign a reduced probability the more distant location.However, if we know that borders represent an impediment to movement (the cost ofwhich we can estimate), that habits play a role, or that target groups are themselvesmobile, then this has implications for behavior, and the mental calculus that motivatesbehavioral habits: movement in some directions will be harder, movement in otherdirections will be easier.

Conventionally, the probability of associated behavior is represented by warping con-tours of space on a map—concave in shape where movement in that direction is hard, andconvex when movement is easy. One novelty of our technique is that it forgoes thiscomplex, ad hoc manipulation, and warps the space instead, so that locations that areeasier to reach (physically, border-crossing wise, or strategically) are embedded closertogether, and locations that are hard to reach are embedded further apart. As a result,concentric circles again demarcate regions that are equally likely to be the site of the next

TERRORISM AND POLITICAL VIOLENCE 19

attack. In other words, a concentric circle drawn on any of the embeddings in this articlerepresents locations at a similar mental strategic distance from its center, given the costassumptions associated with that figure.

Myriad other measures potentially modulate the role of distance in projecting the“next” place. The varying difficulty and desirability of border crossings for migrants intoand within Europe is one example: some countries are attractors and so the distance tothem may seem shorter than they are in the mental calculus of migrants, but some bordersare harder to cross, making distances that involve them seem longer than they are. Theability to incorporate these multiple modulating factors with distance; to create a mentalrepresentation that combines all of these varying criteria into a framework that migrantpopulations might hold and informs their behavioral logic, enables both insight andstrategic reaction in what might otherwise appear as wholly unpredictable settings.

Our approach to political violence gets us closer to a prototype of predictive modelingof conflict of the sort that is widely used by urban police to anticipate crime. Althoughnecessarily somewhat simplistic, it nonetheless lays the groundwork for working towardsbuilding a more comprehensive model of the trade-offs that factor into the selection ofpotential target sets and resource allocation by weighing the costs and benefits of suchdecisions in the light of transaction costs, such as those imposed by borders. Here we havetreated all borders as equally difficult barriers, but the methodology would also allow moredetailed modeling. For example, the efforts by the Mauritanian government to enhanceborder security since 2011 by increasing patrols and working with local tribes make thisborder more difficult to cross, and this could be added into the border-cost matrix torecalibrate the mental landscape, not only of the adjacent countries, but of groups frommuch further afield.

Acknowledgment

This article was made possible in part by a stay at the Hanse-Wissenschaftskolleg Institute forAdvanced Study by Christian Leuprecht and builds in part on Walther O, Skillicorn DB, Zheng Q,Leuprecht C. 2017. Spatial and Temporal Diffusion of Political Violence in North and West Africa,in Walther O, Miles W. (eds). African Border Disorders. Abingdon, Routledge: 87–112.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

David Skillicorn has a Ph.D. from the University of Manitoba and a B.Sc from the University ofSydney. His research focuses and adversarial analytics and human behavior modelling.

Olivier J. Walther is a Visiting Associate Professor in African Studies at the University of Floridaand an Associate Professor in Political Science at the University of Southern Denmark.

Christian Leuprecht is Class of 1965 Professor in Leadership, Royal Military College of Canada,cross-appointed to Queen’s University, and Adjunct Research Professor in the Australian GraduateSchool of Policing and Security at Charles Sturt University.

20 D. B. SKILLICORN ET AL.

Quan Zheng has a Ph.D. from Queen’s University, a Master’s degree in Applied Mathematics fromIndiana University of Pennsylvania and a Master’s degree in Computer Science from University ofUlm.

ORCID

David B. Skillicorn http://orcid.org/0000-0003-0605-4029Olivier Walther http://orcid.org/0000-0002-5768-2845Christian Leuprecht http://orcid.org/0000-0001-9498-4749

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4. Adib Bencherif, “From Resilience to Fragmentation: Al-Qaeda in the Islamic Maghreb andJihadist Group Modularity,” Terrorism and Political Violence (2017): 1–19. Nicolas Desgrais,Yvan Guichaoua, and Andrew Lebovich, “Unity is the Exception. Alliance Formation and De-Formation among Armed Actors in Northern Mali,” Small Wars & Insurgencies 29, no. 4(2018): 654–79.

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19. Catriona Dowd, “Nigeria’s Boko Haram: Local, National and Transnational Dynamics,” inAfrican Border Disorders, ed. Olivier J. Walther and William F.S. Miles (Abingdon: Routledge,2017).

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ACM International Conference on Advances in Social Networks Analysis and Mining(ASONAM), 2015).

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33. Ibid.34. Caitriona Dowd, “Cultural and Religious Demography and Violent Islamist Groups in Africa,”

Political Geography 45 (March 2015): 11–21; Catriona Dowd, “Nigeria’s Boko Haram: Local,National and Transnational Dynamics” (see note 19 above).

35. Denis Retaillé and Olivier Walther, “Conceptualizing The Mobility of Space through theMalian Conflict,” Annales de Géographie 6 (2013).

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